378 HlSTOET OF THE THEOEY OF NUMBERS. [CHAP. XV J. Hermes33 indicated a test for composite Fn by Fermat's theorem. R. Lipschitz34 separated all integers into classes, the primes of one class being Fermat numbers Fn, and placed hi a new light the question of the infinitude of primes Fn. E. Lucas35 stated the result of Proth,23 but with a misprint [Cipolla46]. H. Schemer36 stated that Legendre believed that every Fn is a prime(!), and obtained artificially the factor 641 of F6. He noted (p. 167) that He repeated (pp. 173-8) the test by Pepin,19 with k = 3, and (p. 178) expressed his belief that the numbers (1) are all primes, but had no proof for F16. W. W. R. Ball37 gave references and quoted known results. T. M. Pervouchine38 checked his verification that F12 and F23 are composite by comparing the residues on division by 103— 2. Malvy39 noted that the prime 28+l is not in the series (1). F. Klein40 stated that F7 is composite. A. Hurwitz41 gave a generalization of Proth's23 theorem. Let Fn(x) denote an irreducible factor of degree <£(n) of xn—l. Then if there exists an integer q such that Fp^i(q) is divisible by p, p is a prime. When 1 J. Hadamard42 gave a very simple proof of the second remark by Lucas.21 A. Cunningham43 found that Fn has the factor 319489-974849. A. E. Western44 found that F9 has the factor 216-37+l, F18 the factor 220-13+1, the quotient of F12 by the known factor 214-7-f-l has the factors 216.397+1 and 216-7-139+l. He verified the primality of the factor 241-3+l of F3s, found by J. Cullen and A. Cunningham. He and A. Cunningham found that no more Fn have factors < 106 and similar results. M. Cipolla45 noted that, if q is a prime >(92m~"2-l)/2w+1 and m>l, 2mg4- 1 is a prime if and only if it divides 3*+ 1 for k = g-2m+1. He46 pointed out the misprint in Lucas'35 statement. Nazarevsky47 proved Proth's23 result by using the fact that 3 is a primitive root of a prime 2*+l. 33Arehiv Math. Phys., (2), 4, 1886, 214-5, footnote. 34Jour. fur Math., 105, 1889, 152-6; 106, 1890, 27-29. ^The'orie des nombres, 1891, preface, xii. 36Beitrage zur Zahlentheorie, 1891, 147, 151-2, 155 (bottom), 168. "Math. Recreations and Problems, ed. 2, 1892, 26; ed. 4, 1905, 36-7; ed. 5, 1911, 39-40. 38Math. Papers Chicago Congress of 1893, I, 1896, 277. 39L'interme'diaire des math., 2, 1895, 41 (219). "Vortrage iiber ausgewahlte FragenderElementar Geometric, 1895, 13; French transl., 1896, 26; English transl., "Famous Problems of Elementary Geometry," by Beman and Smith, 1897, 16. "L'interme'diaire des math., 3, 1896, 214. **lbid., p. 114. 43Report British Assoc., 1899, 653-4. The misprint in the second factor has been corrected to agree with the true ** value 2".7.17 + 1. "Cunningham and Western, Proc. Lond. Math. Soc., (2), 1, 1903, 175; Educ. Times, 1903, 270. "Periodico di Mat., 18, 1903, 331. "Also in Annali di Mat., (3), 9, 1904, 141. "L'intermediaire des math., 11, 1904, 215.